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 Sep 3 awarded Promoter Aug 30 comment Looking for references on the complexity of computation of a basis transformation matrix But could there be another approach computing $M$ via solving an equation system? This would sound like possibly reducing the running time to matrix multiplication which would be sufficient. Aug 30 comment Looking for references on the complexity of computation of a basis transformation matrix @ K. Miller: Reducing the running time to the one of matrix multiplication would deliver exactly the desired result: $O^{\sim}(n^\omega d)$. Aug 30 comment Looking for references on the complexity of computation of a basis transformation matrix @K. Miller: I'm looking for an asymptotic running time of $O^{\sim}(n^\omega d)$ where $d$ denotes the maximum of the degrees of the entries of the considered matrix. I'm not doing any implementation, instead I want to reason the asymptotic running time of another algorithm which uses this computation. There is no special structure to exploit. Do you know any references I may consider? We only have the parameters $n$ and $d$ from which the running time should depend on. Aug 30 comment Looking for references on the complexity of computation of a basis transformation matrix Yes, it would be sufficient to be able to compute $M = M_b^{-1}M_a$, but given the matrices $M_a, M_b$, say of degree $d$, computing the inverse $M_b^{-1}$ would be too expensive, see the link in my question. The matrices are defined over $K(x)$, we consider the vector space $V$ of dimension $n$ over $K(x)$. Aug 30 revised Looking for references on the complexity of computation of a basis transformation matrix deleted 1 character in body Aug 30 revised Looking for references on the complexity of computation of a basis transformation matrix added 143 characters in body Aug 30 revised Looking for references on the complexity of computation of a basis transformation matrix added 546 characters in body Aug 30 asked Looking for references on the complexity of computation of a basis transformation matrix Aug 26 asked Asymptotic running time for multiplying multivariate polynomials using Schönhage/Strassen Feb 27 awarded Curious Feb 26 asked Representation Matrix computation time estimation Dec 24 awarded Tumbleweed Nov 12 accepted Prime ideal $P$ in $R$ coprime to the conductor plus the localization $R_{P}$ is a DVR implies that $P$ is invertible Nov 12 comment Prime ideal $P$ in $R$ coprime to the conductor plus the localization $R_{P}$ is a DVR implies that $P$ is invertible The source is part of some lecture notes from my professor and therefore I wasn't sure if I'm allowed to put that here; besides it's written in german. And yes, $P + \mathcal{F} = B$ was used to show that $B_P$ is a DVR. Again, thank you very much! Nov 12 comment Prime ideal $P$ in $R$ coprime to the conductor plus the localization $R_{P}$ is a DVR implies that $P$ is invertible I thought about this before but I kind of scrapped that idea for no reason. Of course this is the answer, it's clear now! Thank you very much. Nov 12 asked Prime ideal $P$ in $R$ coprime to the conductor plus the localization $R_{P}$ is a DVR implies that $P$ is invertible Jul 31 awarded Teacher Jul 25 comment Why is every Divisor of the rational function field $K(x)$ over $K$ a principal divisor if $K$ is algebraically closed So the statement is wrong as I suggested. Because the degree of a prime divisor is 1. Jul 24 asked Why is every Divisor of the rational function field $K(x)$ over $K$ a principal divisor if $K$ is algebraically closed